Les écoulements rampants, recirculants et instationnaires d'un fluide viscoplastique autour d'un cylindre ont été étudiés.Numériquement, les morphologies des écoulements, la localisation des zones rigides, les champs de contraintes et pression autour du cylindre ainsi que le coefficient de traînée, ont été déterminés sur un large domaine des nombres de Reynolds et d'Oldroyd.Expérimentalement, les fluides étudiés sont des gels de polymère Carbopol®. Le comportement élastoviscoplastique de ces gels a été modélisé par une loi d'Herschel-Bulkley adaptée. Le montage expérimental conçu et réalisé a été validé par l'étude de l'écoulement d'un fluide newtonien autour d'un cylindre et la mise en place d'une procédure adaptée pour les fluides à seuil.On a pu constater l'influence des conditions d'interface avec l'apparition d'une morphologie de lâchers de tourbillons simultanés et symétriques.

Escoamentos sem inércia de fluido elasto-viscoplástico, dentro de uma cavidade, são numericamente analisados. As soluções visam compreender a influência dos efeitos viscosos e elásticos na topologia de superfícies de escoamento. Assumindo-se que o colapso da microestrutura do material é instantâneo, o modelo mecânico é constituído pelas equações governantes de massa e momentum para fluidos incompressíveis, associado a uma equação hiperbólica para o tensor tensão extra, baseado na equação do modelo Oldroyd-B (Nassar et al, 2011). A principal característica do modelo é considerar a viscosidade e o tempo de relaxação como função da taxa de deformação, permitindo a pseudoplasticidade de viscosidade e restringindo os efeitos elásticos para as regiões não deformadas do material. As simulações numéricas são realizadas através do método de Galerkin mínimos quadrados a três campos: tensor tensão extra, pressão e velocidade. Os resultados mostram que as superfícies de escoamento do material são fortemente influenciadas pela ação combinada entre os efeitos elásticos e viscosos, estando em conformidade com a recente visualização experimental dos fluxos elasto-viscoplásticos. / Elasto viscoplastic uid ows without inertia, within a cavity, are numerically analyzed. The solutions aim to understand the in uence of viscous and elastic e ects on the topology of yield surfaces. Assuming that the collapse of the material microstructure is instantaneous, the mechanical model consists of the governing equations of mass and momentum for incompressible uids, associated with a hyperbolic equation for the extra stress tensor, based on the equation of the Oldroyd-B model (Nassar et al, 2011). The main feature of model is to consider the viscosity and the relaxation time as a function of shear rate, allowing the shear-thinning of viscosity and restricting elastic e ects for regions not deformed material. Numerical simulations are performed by the method of Galerkin Least Squares to three elds: extra stress tensor, pressure and velocity. The results show that the yield surfaces of material are strongly in uenced by the combined action between the elastic and viscous e ects, complying with the recent experimental visualization of elasto-viscoplastic ows.

Escoamentos sem inércia de fluido elasto-viscoplástico, dentro de uma cavidade, são numericamente analisados. As soluções visam compreender a influência dos efeitos viscosos e elásticos na topologia de superfícies de escoamento. Assumindo-se que o colapso da microestrutura do material é instantâneo, o modelo mecânico é constituído pelas equações governantes de massa e momentum para fluidos incompressíveis, associado a uma equação hiperbólica para o tensor tensão extra, baseado na equação do modelo Oldroyd-B (Nassar et al, 2011). A principal característica do modelo é considerar a viscosidade e o tempo de relaxação como função da taxa de deformação, permitindo a pseudoplasticidade de viscosidade e restringindo os efeitos elásticos para as regiões não deformadas do material. As simulações numéricas são realizadas através do método de Galerkin mínimos quadrados a três campos: tensor tensão extra, pressão e velocidade. Os resultados mostram que as superfícies de escoamento do material são fortemente influenciadas pela ação combinada entre os efeitos elásticos e viscosos, estando em conformidade com a recente visualização experimental dos fluxos elasto-viscoplásticos. / Elasto viscoplastic uid ows without inertia, within a cavity, are numerically analyzed. The solutions aim to understand the in uence of viscous and elastic e ects on the topology of yield surfaces. Assuming that the collapse of the material microstructure is instantaneous, the mechanical model consists of the governing equations of mass and momentum for incompressible uids, associated with a hyperbolic equation for the extra stress tensor, based on the equation of the Oldroyd-B model (Nassar et al, 2011). The main feature of model is to consider the viscosity and the relaxation time as a function of shear rate, allowing the shear-thinning of viscosity and restricting elastic e ects for regions not deformed material. Numerical simulations are performed by the method of Galerkin Least Squares to three elds: extra stress tensor, pressure and velocity. The results show that the yield surfaces of material are strongly in uenced by the combined action between the elastic and viscous e ects, complying with the recent experimental visualization of elasto-viscoplastic ows.

Escoamentos sem inércia de fluido elasto-viscoplástico, dentro de uma cavidade, são numericamente analisados. As soluções visam compreender a influência dos efeitos viscosos e elásticos na topologia de superfícies de escoamento. Assumindo-se que o colapso da microestrutura do material é instantâneo, o modelo mecânico é constituído pelas equações governantes de massa e momentum para fluidos incompressíveis, associado a uma equação hiperbólica para o tensor tensão extra, baseado na equação do modelo Oldroyd-B (Nassar et al, 2011). A principal característica do modelo é considerar a viscosidade e o tempo de relaxação como função da taxa de deformação, permitindo a pseudoplasticidade de viscosidade e restringindo os efeitos elásticos para as regiões não deformadas do material. As simulações numéricas são realizadas através do método de Galerkin mínimos quadrados a três campos: tensor tensão extra, pressão e velocidade. Os resultados mostram que as superfícies de escoamento do material são fortemente influenciadas pela ação combinada entre os efeitos elásticos e viscosos, estando em conformidade com a recente visualização experimental dos fluxos elasto-viscoplásticos. / Elasto viscoplastic uid ows without inertia, within a cavity, are numerically analyzed. The solutions aim to understand the in uence of viscous and elastic e ects on the topology of yield surfaces. Assuming that the collapse of the material microstructure is instantaneous, the mechanical model consists of the governing equations of mass and momentum for incompressible uids, associated with a hyperbolic equation for the extra stress tensor, based on the equation of the Oldroyd-B model (Nassar et al, 2011). The main feature of model is to consider the viscosity and the relaxation time as a function of shear rate, allowing the shear-thinning of viscosity and restricting elastic e ects for regions not deformed material. Numerical simulations are performed by the method of Galerkin Least Squares to three elds: extra stress tensor, pressure and velocity. The results show that the yield surfaces of material are strongly in uenced by the combined action between the elastic and viscous e ects, complying with the recent experimental visualization of elasto-viscoplastic ows.

We prove a global-in-time and large-data existence of a suitable weak solution to a system of partial differential equations describing an unsteady flow of homogeneous incom- pressible viscoelastic rate-type fluid. The material parameters are continuous functions of temperature and, in particular, the dependence of the shear modulus is assumed to be linear. It is shown that studied models obey the fundamental laws of thermodynamics. The key step towards the existence proof is derivation of the balance of entropy. This in- equality is paramount in the analysis and as its consequence, we obtain sufficient a priori estimates, positivity of temperature and also regularity of the elastic deformation. The second part of the thesis deals with the existence analysis for the isothermal case, however using a completely different method, which is of independent interest. 1

Les polymères jouent un rôle important sur la réduction de la traînée et la modification de la structure des écoulements. Nous avons utilisé le modèle Oldroyd-B pour étudier l’effet de la viscoélasticité de solutions de polymères dilués sur des écoulements en deux dimensions autour d’obstacles dans un canal. Les obstacles sont pris en compte par la méthode de pénalisation volumique et des condition aux limites artificielles sans réflexion sont imposées à la sortie du canal. La discrétisation est effectuée par des schémas performants en différences finies et la résolution par une méthode multigrille. Les simulations numériques sont effectuées pour une large gamme de nombres de Reynolds et de nombres de Weissenberg. Les caractéristiques détaillées des écoulements viscoélastiques sont analysées et comparées entre elles et celles de l’écoulement du fluide sans polymère. En particulier les jeux de paramètres conduisant à une augmentation ou à une baisse de la traînée. Enfin les effets du polymère sur des écoulements turbulents sont aussi analysés. / Polymer plays an important role on the drag reduction and modification of the structure of flow. In this thesis, Oldroyd B model is employed to study the effectof viscoelasticity for polymer solutions diluted by two-dimensional direct numerical simulation. The obstacles are taken into account by penalization method. The artificial boundary condition is imposed without any reflection on the channel outlet. Flow pasta cylinder is investigated in detailed by present numerical methods. The numerical codes are valid for predicting the drag force and capturing the important character of viscoelastic flow by comparing with experimental and other numerical results. The drag map of the cylinder is obtained at a wide range of Reynolds number and Weissenberg number space. The detailed characteristics of viscoelastic flow are reported in the thesis. The effects of polymer on two-dimensional turbulent flow are also discussed by grid turbulent flow.

La première contribution de cette thèse est l'étude de la modélisation ainsi que de la discrétisation des écoulements multiphasiques en géométrie microfluidique. Nous proposons un nouveau schéma d'ordre deux de discrétisation basé sur les méthodes mixtes volumes finis/éléments finis pour le système de Stokes bi-fluides avec tension de surface. Nous présentons alors des comparaison de la précision de ce nouveau schéma avec la discrétisation MAC, en 2D et en 3D-axi. La seconde contribution est relative à l'étude de schémas numériques en temps pour les fluides viscoélastiques. Nous présentons les limites actuelles des modélisations dans ce domaine en étudiant le cas des écoulements de micelles géantes, polymères ayant la capacité de se réorganiser spatialement en fonction du taux de cisaillement. Nous montrons qu'une condition de stabilité liée au ratio de viscosité du polymère et du solvant en temps - très restrictive - existe. Un nouveau schéma est alors proposé pour contourner cette limitation et des études stabilité sont menés pour démontrer nos résultats.

Recently, a new kind of turbulence has been discovered in the flow of concentrated polymer melts and solutions. These flows, known as purely elastic flows, become unstable when the elastic forces are stronger than the viscous forces. This contrasts with Newtonian turbulence, a more familiar regime where the fluid inertia dominates. While there is little understanding of purely elastic turbulence, there is a well-established dynamical systems approach to the transition from laminar flow to Newtonian turbulence. In this project, I apply this approach to purely elastic flows. Laminar flows are characterised by ordered, locally-parallel streamlines of fluid, with only diffusive mixing perpendicular to the flow direction. In contrast, turbulent flows are in a state of continuous instability: tiny differences in the location of fluid elements upstream make a large difference to their later locations downstream. The emerging understanding of the transition from a laminar to turbulent flow is in terms of exact coherent structures (ECS) — patterns of the flow that occur near to the transition to turbulence. The problem I address in this thesis is how to predict when a purely elastic flow will become unstable and when it will transition to turbulence. I consider a variety of flows and examine the purely elastic instabilities that arise. This prepares the ground for the identification of a three-dimensional steady state solution to the equations, corresponding to an exact coherent structure. I have organised my research primarily around obtaining a purely elastic exact coherent structure, however, solving this problem requires a very accurate prediction of the exact solution to the equations of motion. In Chapter 2 I start from a Newtonian ECS (travelling wave solutions in two-dimensional flow) and attempt to connect it to the purely elastic regime. Although I found no such connection, the results corroborate other evidence on the effect of elasticity on travelling waves in Poiseuille flow. The Newtonian plane Couette ECS is sustained by the Kelvin-Helmholtz instability. I discover a purely elastic counterpart of this mechanism in Chapter 3, and explore the non-linear evolution of this instability in Chapter 4. In Chapter 5 I turn to a slightly different problem, a (previously unexplained) instability in a purely elastic oscillatory shear flow. My numerical analysis supports the experimental evidence for instability of this flow, and relates it to the instability described in Chapter 3. In Chapter 6 I discover a self-sustaining flow, and discuss how it may lead to a purely elastic 3D exact coherent structure.

Les écoulements rampants, recirculants et instationnaires d’un fluide viscoplastique autour d’un cylindre ont été étudiés.Numériquement, les morphologies des écoulements, la localisation des zones rigides, les champs de contraintes et pression autour du cylindre ainsi que le coefficient de traînée, ont été déterminés sur un large domaine des nombres de Reynolds et d’Oldroyd.Expérimentalement, les fluides étudiés sont des gels de polymère Carbopol®. Le comportement élastoviscoplastique de ces gels a été modélisé par une loi d’Herschel-Bulkley adaptée. Le montage expérimental conçu et réalisé a été validé par l'étude de l'écoulement d'un fluide newtonien autour d'un cylindre et la mise en place d’une procédure adaptée pour les fluides à seuil.On a pu constater l'influence des conditions d’interface avec l’apparition d’une morphologie de lâchers de tourbillons simultanés et symétriques. / Creeping, recirculating and unsteady flows around a cylinder for yield stress fluid were studied.Numerically, morphologies of the flows, the location of the unyielded zones, the pressure and stress fields around the cylinder and the drag coefficient were determined over a wide range of Reynolds and Oldroyd numbers.Experimentally, fluids studied are polymer gels Carbopol®. The elastoviscoplastic behavior of these gels was modeled by a Herschel-Bulkley adapted law. The experimental setup was designed and validated by studying the flow of a newtonian fluid around a cylinder. An appropriate procedure for the viscoplastic fluid was implemented.We observe the influence of interface conditions with the appearance of a morphology showing simultaneous and symmetrical vortex shedding.

Cette thèse est consacrée à la vérification de l'analogie entre l'instabilité viscoélastique (VEI) et l'instabilité magnéto-rotationnel (MRI) dans un écoulement képlérien, afin de mieux comprendre le transport du moment dans les disques d'accrétion. Le discriminant de Rayleigh élasto-rotationnel est établi pour clarifier le rôle de l'élasticité dans le VEI. L'analyse de stabilité linéaire (LSA) avec le modèle d’Oldroyd-B est effectuée pour prédire les paramètres critiques des modes viscoélastiques. Il fait apparaître également l'influence de l'élasticité, la viscosité polymérique et d'autres paramètres de contrôle pour le VEI. Des expériences bien contrôlées avec des solutions aqueuses de polyoxyéthylène (POE) et de polyéthylène glycol (PEG) sont effectuées. Nous avons observé le mode stationnaire axisymétrique supercritique avec des solutions de faible élasticité et modes désordonnés sous-critiques avec des solutions de grande élasticité. Les formes et les valeurs critiques de ces modes sont en bon accord avec les prédictions théoriques de LSA. Selon l'analogie, le mode axisymétrique stationnaire est probablement l'analogue de MRI standard, tandis que le mode désordonné est probable que l'analogue de MRI hélicoïdale. La thèse contient aussi des résultats théoriques expérimentaux sur quatre autres régimes de rotation et un cas de limite d'élasticité infinie. / This thesis is devoted to the verification of the analogy between the viscoelastic instability (VEI) and the magnetorotational instability (MRI) in a Keplerian flow, in order to get better understanding of the momentum transportation in accretion disks.The elasto-rotational Rayleigh discriminant is deduced to clarify the role of the elasticity in the VEI. The linear stability analysis (LSA) with Oldroyd-B model is performed to predict critical parameters of viscoelastic modes, and it reveals the influence of the elasticity, polymer viscosity on the VEI. Experiments with well controlled aqueous solutions of polyoxyethylene (POE) and polyethylene glycol (PEG) are conducted. We have observed supercritical stationary axisymmetric mode with solutions of small elasticity and subcritical disordered modes with solutions of large elasticity. Both the flow patterns and the critical values of these modes are in good agreement with the LSA predictions. According to the analogy, the stationary axisymmetric mode is likely the analog of the standard MRI while the disordered mode is likely the analog of the helical MRI. The thesis contains also theoretical and experimental results with four other rotation regimes and the limit case of infinite elasticity.